On the homotopy type of the Deligne–Mumford compactification

Ebert, Johannes; Giansiracusa, Jeffrey

doi:10.2140/agt.2008.8.2049

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On the homotopy type of the Deligne–Mumford compactification

Johannes Ebert and Jeffrey Giansiracusa

Algebraic & Geometric Topology 8 (2008) 2049–2062

DOI: 10.2140/agt.2008.8.2049

Abstract

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne–Mumford compactification. We give an integral refinement: the classifying space of the Charney–Lee category actually has the same homotopy type as the moduli stack of stable curves, and the étale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney–Lee category.

Keywords

Deligne–Mumford compactification, moduli of curves, stack, mapping class group, orbit category

Mathematical Subject Classification 2000

Primary: 32G15

Secondary: 30F60, 14A20, 14D22

References

Publication

Received: 16 July 2008

Accepted: 24 September 2008

Published: 5 November 2008

Authors

Johannes Ebert
Mathematisches Institut der Universität Bonn Beringstrasse 1 53115 Bonn Germany

Jeffrey Giansiracusa
Mathematical Institute University of Oxford 24–29 St. Giles’ Oxford, OX1 3LB United Kingdom

Volume 8, issue 4 (2008)